Antilog Calculator – Antilogarithm

Find the inverse of a logarithm

Other Titles
Exploring Antilogarithms: The Inverse of Logarithms
A complete guide to understanding what antilogarithms are, how to calculate them, and where they are used in the real world.

Understanding Antilog Calculator: A Comprehensive Guide

  • What is an antilogarithm?
  • The relationship between logs and antilogs
  • The formula: x = antilog_b(y) = b^y
An antilogarithm, or 'antilog', is the inverse operation of a logarithm. Just as division undoes multiplication, an antilog undoes a logarithm. If you have the logarithm of a number, the antilog allows you to find the original number.
The core relationship is simple: if logb(x) = y, then the antilogarithm is x = antilogb(y). This is mathematically equivalent to the exponential form x = b^y.
In essence, calculating the antilogarithm is the same as performing exponentiation. You are raising the logarithm's base (b) to the power of the logarithm's value (y) to get back the original number (x).
Common Bases

Fundamental Examples

  • If log₁₀(100) = 2, then antilog₁₀(2) = 10² = 100.
  • If ln(7.389) ≈ 2, then antilog_e(2) = e² ≈ 7.389.
  • If log₂(8) = 3, then antilog₂(3) = 2³ = 8.

Step-by-Step Guide to Using the Antilog Calculator

  • How to enter the base and value
  • Using common bases like 10 and e
  • Interpreting the calculated result
Our calculator makes finding the antilog straightforward. Here's how to use it:
Input Fields
Calculation
Click the 'Calculate' button. The calculator computes b^y to find the antilogarithm 'x'. The result is the number you would have started with to get the logarithm 'y'.

Usage Scenarios

  • To find the antilog of 3 in base 10: Enter Base (b) = 10, Value (y) = 3. Result: 1000.
  • To find the natural antilog of 2.5: Enter Base (b) = 2.71828, Value (y) = 2.5. Result: ≈12.18.
  • To find antilog₂(5): Enter Base (b) = 2, Value (y) = 5. Result: 32.

Real-World Applications of Antilog Calculations

  • Reversing logarithmic scales in science
  • Applications in chemistry, seismology, and acoustics
  • Financial growth calculations
Antilogarithms are crucial for converting data from a logarithmic scale back to a linear scale. This is vital in fields that use logarithmic scales to handle numbers spanning many orders of magnitude.
Chemistry: pH Scale
The pH of a solution is defined as pH = -log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration. To find the [H⁺] from a known pH value, you use the antilog: [H⁺] = 10^(-pH).
Seismology: Richter Scale
The Richter scale is logarithmic. The magnitude M is related to the seismograph's wave amplitude A by M = log₁₀(A/A₀). To find how much stronger one earthquake is than another, you compare their amplitudes, which requires using antilogs to find A from M.
Acoustics: Decibel (dB) Scale
The loudness of sound in decibels is L = 10 log₁₀(I/I₀), where I is the sound intensity. To find the intensity 'I' of a sound from its dB level, you must calculate I = I₀ 10^(L/10), which involves an antilog calculation.

Practical Examples

  • If a solution has a pH of 4, the hydrogen ion concentration is [H⁺] = 10⁻⁴ = 0.0001 mol/L.
  • An earthquake of magnitude 7 is 10 times more powerful in wave amplitude than one of magnitude 6, because 10⁷ is 10 times 10⁶.
  • The intensity of a 60 dB sound is 10 times greater than a 50 dB sound.

Common Misconceptions and Correct Methods in Antilogarithms

  • Confusing antilog with 1/log
  • Using the wrong base
  • The importance of the base
Misconception 1: Antilog is the Reciprocal
A common error is thinking antilogb(y) is the same as 1 / logb(y). This is incorrect. Antilog is the inverse function (exponentiation), not the multiplicative inverse (reciprocal).
Misconception 2: Forgetting the Base
The term 'antilog' is meaningless without a base. antilog(y) could mean 10^y, e^y, or 2^y. The base is critical. If a problem just says 'log', it usually implies base 10. If it says 'ln', it means base e.
The Correct Method
Always identify the base 'b' and the logarithm value 'y'. The calculation is always b^y. This simple rule is the foundation of all antilog calculations. Our calculator requires you to be explicit about the base to avoid confusion.

Correct vs. Incorrect

  • Correct: antilog₁₀(2) = 10² = 100.
  • Incorrect: antilog₁₀(2) is NOT 1 / log₁₀(2).

Mathematical Derivation and Examples

  • The formal inverse relationship
  • Deriving the exponential form
  • Worked examples
Formal Definition of Inverse Functions
Let f(x) = logb(x). The inverse function, f⁻¹(y), is the antilogarithm. By definition of inverse functions, f(f⁻¹(y)) = y. Substituting f(x), we get logb(f⁻¹(y)) = y.
To solve for f⁻¹(y), we convert this logarithmic equation to its exponential form. The base is 'b', the exponent is 'y', and the result is the expression inside the logarithm, f⁻¹(y). Therefore, f⁻¹(y) = b^y.
This confirms that the antilogarithm function is the exponential function: antilog_b(y) = b^y.

Detailed Worked Example: Find antilog_4(2.5)

  • 1. Identify the base and value: Base (b) = 4, Value (y) = 2.5.
  • 2. Set up the exponential equation: x = 4^2.5.
  • 3. Calculate the power: 4^2.5 can be written as 4^(5/2) or (√4)⁵.
  • 4. Step 1: Calculate the square root of 4, which is 2.
  • 5. Step 2: Raise the result to the power of 5: 2⁵ = 32.
  • Solution: antilog_4(2.5) = 32.